2 Answers
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There is no international authority that sets any such limit, nor one which would have the ability to enforce it.

But for most people, it is not practical to maintain too many multiple nationalities. Most countries frown upon multiple citizenships, and place some restrictions on it, at least on paper. Some countries, like Austria, only allow dual citizenship from birth, others, like Japan, do not recognize the concept at all, and will force you to choose one or relinquish your Japanese citizenship if they find out that you have others.

In other cases, retaining a nationality for a country where you do not reside may not be worth the trouble. The United States, for example, attempts to collect tax on worldwide income, whereas most countries only tax income earned within that country. Enforcement of this policy has become far stricter in recent years, and some U.S. expatriates are choosing to give up their U.S. citizenship rather than have to file tax returns to a country where they may not have resided for decades.

There is no limit, in theory or practice. This thread on Flyertalk mentions several people with five citizenships, posits a case where a child born to a couple with three citizenships each would easily get 7, and goes on to speculate that with the right sets of parents it would be possible to acquire more than 9 at birth plus any more you'd care to accumulate during your lifetime. There are still quite a few countries where you can become a citizen in less than 5 years, or even buy citizenship outright if you throw enough money at them.

Once past the age of 20 or so, simply maintaining a citizenship generally requires zero effort, once the bit is toggled it stays toggled short of (say) actively waging war against a country you're a citizen of.

However, citizenship does often incur obligations like tax, military service etc, which can make it a pain to stay in the good books of all your countries. As a simple example, if I'm a male Finnish citizen over 18 who hasn't completed their national service, I'll stay a citizen if I stay overseas, but I'll be arrested as a draft dodger if I return to the country.